Analytical and Machine Learning Study of One-Dimensional Non-Interacting Spinless Trapped Fermionic Systems - Ground-State Properties

Aug 4, 2026

Study of ground-state properties in three one-dimensional, non-interacting, fully polarized (spinless) fermionic systems of N identical particles confined in different trapping potentials. For harmonic, infinite well, and Morse potentials, the ground-state wavefunction is expressed via a Vandermonde determinant. One-body density matrices and pair correlation functions are computed and shown to match using two analytical methods, then provided in closed polynomial forms for general N. Results are further reproduced and validated with a machine-learning Neural Quantum State trained by variational Monte-Carlo, confirming energies and density matrices.

LanguageEnglish

CategoryResearch & Report

FormatPDF

Pages48

Reading Time2 h 1 min

AccessFree to download

Introduction

Mathematical study of one-dimensional quantum systems

Vandermonde determinants

Many-body density matrices

Harmonic oscillator

Infinite well

Morse oscillator

Study of the ground state using Machine Learning

Artificial Neural Networks

Methodology

Comparison with the analytical results

Conclusions

Bibliography

Derivation of useful formulae

Derivations for the harmonic oscillator

Closed expressions for the infinite well OBDM and PCF

Derivations for the Morse oscillator

Ground-state calculations

Plots of the OBDM and the PCF obtained with ML

Analytical and Machine Learning study of onedimensional non-interacting spinless trapped fermionic systems Jaume Rius Casado Supervised by: Arnau Rios Huguet Departament de Física Quàntica i Astrofísica, Universitat de Barcelona (UB), E-08028 Barcelona, Spain 28 August 2023 In this work we study the ground-state properties of three different onedimensional systems of N identical, non-interacting, spinless fermions trapped in a potential well. We consider a harmonic trap, an infinite potential well anda Morse potential, and for all of them we prove that the ground-state wavefunction can be written in terms of a Vandermonde determinant. We compute and plot the one-body density matrix and the pair correlation function for systems of 2 to 5 particles using two different analytical methods to check that both provide the same results. Moreover, we derive closed expressions for these functions in terms of polynomials for a general number of particles. These polynomials, in turn, can be expressed using Vandermonde vectors and square matrices. To complement the mathematical study of the systems, we reproduce and validate the analytical results using a Machine Learning approach. We use a Neural Quantum State as an ansatz for the ground-state wavefunction anda Variational Monte-Carlo method to find the best neural network parameters. Both the energies and the density matrices are correctly reproduced using Machine Learning. Keywords: Vandermonde determinant, particles in a trap, ground state, one-body density matrix, pair correlation function, Hermite polynomials, Laguerre polynomials, Neural Quantum State. Acknowledgements I would like to express my gratitude to my advisor Dr. Arnau Rios Huguet for his continuous guidance and immense support throughout this project. Also, I would like to thank my friend Javier Rozalén Sarmiento, who has always been available and ready to answer any question I had about Machine Learning. Jaume Rius Casado: [jriuscas8@alumnes.ub.edu](jriuscas8@alumnes.ub.edu) Contents 1 Introduction 3 2 Mathematical study of one-dimensional quantum systems 4 2.1 Vandermonde determinants ........................... 4 2.2 Many-body density matrices ........................... 5 2.2.1 Integral formulae for the density matrices ............... 5 2.2.2 Uncorrelated formulae for the OBDM and the PCF .......... 6 2.3 Harmonic oscillator ................................ 6 2.3.1 Description and ground-state wavefunction .............. 6 2.3.2 Plots of the OBDM and the PCF .................... 8 2.3.3 Closed expressions for the OBDM and the PCF ............ 9 2.4 Infinite well .................................... 11 2.4.1 Description and ground-state wavefunction .............. 11 2.4.2 Plots of the OBDM and the PCF .................... 12 2.4.3 Closed expressions for the OBDM and the PCF ............ 14 2.5 Morse oscillator .................................. 15 2.5.1 Description and ground-state wavefunction .............. 15 2.5.2 Plots of the OBDM and PCF ...................... 17 2.5.3 Closed expressions for the OBDM and the PCF ............ 18 2.6 Discussion and comparison of the different plots ................ 18 3 Study of the ground state using Machine Learning 19 3.1 Artificial Neural Networks ............................ 19 3.2 Methodology ................................... 20 3.3 Comparison with the analytical results ..................... 21 4 Conclusions 22 Bibliography 23 A Derivation of useful formulae 25 A. 1 Vandermonde determinant formula . . . . . . . . . . . . . . . . . . . . . . . 25 A.2 Uncorrelated formulae for the OBDM and the PCF . . . . . . . . . . . . . . 26 B Derivations for the harmonic oscillator 30 B.1 Slater determinant of Hermite polynomials as a VD .............. 30 B.2 Closed expressions for the harmonic oscillator OBDM and PCF . . . . . . . 32 C Closed expressions for the infinite well OBDM and PCF 35 D Deriva

FAQs

What types of one-dimensional fermionic systems are analyzed in this study?

The work considers N identical, non-interacting, spinless (fully polarized) fermions confined in a potential well, specifically using a harmonic trap, an infinite potential well, and a Morse potential.

How are the ground-state wavefunctions represented for these trapped systems?

For each trapping potential, the ground-state wavefunction is proven to be expressible using a Vandermonde determinant.

How does machine learning contribute to validating the analytical results?

A Neural Quantum State ansatz combined with a variational Monte-Carlo method is used to optimize neural network parameters, and the model reproduces energies and density matrices consistent with analytical computations.

Aug 4, 2026

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